Finding functions that satisfy a condition

Question

Let $(a_n)_n$ be a positive numbers sequence such that$$ \lim_{n\to\infty} a_n =0 $$ Find the functions $f :R \to R$ that admit primitives such that $$2f(x)=f(x+a_n)+f(x-a_n)$$ for every x $\in R$ and n $\in N$
I've been struggling for a while with this problem. I've noticed that the linear functions satisfy the conditions but I don't know how I might prove that there aren't other functions. My instinct was to first prove that $f$ is monotonous, since the existence of primitives ensures that the function has Darboux, but I didn't have any success.